Mathematics in the Modern World

Mathematics - is a branch of science, which deals with numbers and their

operations.
- It involves calculations, computation, solving of problems, etc.
- helps us to organize and systemize our ideas about patterns, in so doing,
not only can we admire and enjoy these patterns, we can also use them to
infer some of the underlying principles that govern the world of nature.
Patterns and Numbers in Nature:
“Mathematics is a study of patterns and relationship, a way of thinking, an art, a
language, and a tool. It is about patterns and relationships. Numbers are just a
way to express those patterns and relationships.” - National Council of Teachers
of Mathematics (1991)
Pattern - is an arrangement which helps observers anticipate what they might see
or what happens next.
- shows what may have come before.

Natural patterns - include symmetries, fractals, spirals, meanders, waves, foams,

tessellations, cracks, and stripes. Studying patterns allows one to watch, guess,
create, and discover. The present mathematics is considerably more than
arithmetic, algebra, and geometry.
Natural Patterns:
1. TESSELLATION - is a pattern of one or more shapes where the shapes do
not overlap or no have space between them.

2. Waves - in water or wind pass over sand, they create patterns of ripples.
When winds blow over large bodies of sand, they create dunes, sometimes
in extensive

3. Fractals - are never-ending patterns. The beauty of fractals is that their

infinite complexity is formed through the repetition of simple equations.
These repeating patterns are displayed at every scale.
- is a kind of pattern that we observe often in nature and in art. As Ben
Weiss explains, “whenever you observe a series of patterns repeating
over and over again, at many different scales, and where any small
part resembles the whole, that’s a fractal.

4. Spiral - is a curved pattern that focuses on a center point and a series of

circular shapes that revolve around it. Examples of spirals are pine cones,
pineapples, hurricanes. The reason for why plants use a spiral form like the
leaf picture below is because they are constantly trying to grow but stay
secure.

5. CHAOS, FLOW, MEANDERS - The relationship between chaos and fractals is

that strange attractors in chaotic systems have a fractal dimension. ...
Meanders are bends in a sinuous form that appears as rivers or other
 channels, which form as a fluid, most often water, flows around bends.
Chaos is the study of how simple patterns can be generated from
complicated underlying behavior. Many events were considered to be
chaotic, unpredictable and random. The dripping of a tap, the weather, the
formation of clouds, the fibrillation of the human heart, the turbulence of
fluid flows or the movement of a simple pendulum under the influence of a
number of magnets are a few examples.

6. SPOTS, STRIPES - Leopards and ladybirds are spotted; angelfish and zebras
are striped.
- These patterns have an evolutionary explanation: they have
functions which increase the chances that the offspring of the
patterned animal will survive to reproduce.
- One function of animal patterns is camouflage; for instance, a
leopard that is harder to see catches more prey.

7. Cracks - are linear openings that form in materials to relieve stress. When a
material fails in all directions it results in cracks. The patterns created reveal
if the material is elastic or not.
- are overlooked because they are so common. It is often a pattern
engineers want to avoid, for example a crack in a bridge or a road or
a glass. Engineers spend a lot of time trying to determine when a
crack can become a catastrophe.

8. Foam - is a mass of bubbles; foams of different materials occur in nature

 - is a substance made by trapping air or gas bubbles inside a solid or
liquid. Typically, the volume of gas is much larger than that of the
liquid or solid, with thin films separating gas pockets.
Bubble - is a spherically contained volume of air or other gas, especially one made
from soapy liquid while foam is a substance composed of a large collection of
bubbles or their solidified remains.

Symmetry - can be found everywhere. It can be seen from different viewpoints

namely; nature, the arts and architecture, mathematics; especially geometry and
science.
- occurs when there is congruence in dimensions, due proportions and
arrangement. It provides a sense of harmony and balance.
- in everyday language refers to a sense of harmonious and beautiful
proportion and balance. In mathematics, "symmetry" has a more precise
definition, and is usually used to refer to an object that is invariant under
some transformations; including translation, reflection, rotation or scaling.
Although these two meanings of "symmetry" can sometimes be told apart,
they are intricately related.

TYPES OF SYMMETRY:
1. Bilateral or reflection symmetry - is the simplest kind of symmetry. It can
also be called mirror symmetry because an object with this symmetry looks
unchanged if a mirror passes through its middle.
2. Radial symmetry - is rotational symmetry around a fixed point known as the
center. Images with more than one lines of symmetry meeting at a
common point exhibits a radial symmetry.

3. Rosette patterns - consist of taking motif or an element and rotating and/or

reflecting that element. There are two types of rosette patterns namely
cyclic and dihedral.

4. Frieze pattern - is a pattern in which a basic motif repeats itself over and
over in one direction. It extends to the left and right in a way that the
pattern can be mapped onto itself by a horizontal translation.
A. Hop - only admits a translational symmetry.

B. Step - only admits a translational and glide symmetries.

C. Sidle - only admits translations and vertical reflections.

D. Spinning Hop - only admits translations and 180◦ rotations (half-turns).

E. Spinning Sidle - only admits translations, vertical reflections, rotations,

and glide reflections.
 F. Jump - only admits translations, a horizontal reflection, and glide
reflection.

G. Spinning Jump - admits translations, vertical reflections, horizontal

reflections, rotations, and glide reflections.

5. Wallpaper pattern - is a pattern with translation symmetry in two

directions. It is, therefore, essentially an arrangement of friezes stacked
upon one another to fill the entire plane.

Fibonacci Sequence:
The Fibonacci sequence - was invented by the Italian Leonardo Pisano Bigollo
(1180-1250), who is known in mathematical history by several names: Leonardo
of Pisa (Pisano means “from Pisa”) and Fibonacci (which means “son of Bonacci”).
One of the exercises in Fibonacci’s book:
“A man put a pair of rabbits in a place surrounded on all sides by a wall. How
many pairs of rabbits are produced from that pair in a year, if it is supposed that
every month each pair produces a new pair, which from the second month
onwards becomes productive?”
Rabbit Habit:

Luca Pacioli - found the relationship between Fibonacci sequence and the golden
ratio.
Golden ratio - was first called as the Divine Proportion in the early 1500s in
Leonardo da Vinci’s work was explored by Luca Pacioli (Italian mathematician)
entitled “De Devina Proportione” in 1509.
- We see that n gets larger and larger, the ratio gets closer and closer to a
value denoted by ℓ is called the golden ratio
- Golden Ratio is the relationship between numbers on the Fibonacci
sequence where plotting the relationships on scales results in a spiral shape

 Da Vinci’s drawings of the five platonic solids and it was probably da Vinci
who first called it the “section aurea” Latin for Golden Section
 Two quantities are in the Golden ratio if their ratio is the same of their sum
to the larger of the two quantities.
 The Fibonacci numbers can be applied to the Golden rectangle
Golden rectangle - proportions of a rectangle
- is known as one of the most visually satisfying of all geometric forms –
hence, the appearance of the Golden ratio in art.
 - is also related to the Golden spiral, which is created by making adjacent
squares of Fibonacci dimensions.
- can be broken into squares the size of the next Fibonacci number down and
below.
Fibonacci spiral - which approximates the golden spiral, using Fibonacci sequence
square sizes up to 34.
- Take a golden rectangle, break it down into smaller squares based from
Fibonacci sequence and divide each with an arc.

Fibonacci numbers appear in nature in various places:

 Pinecones, Speed Heads, Vegetables and Fruits Spiral patterns curving from
left and right can be seen at the array of seeds in the center of a sunflower.
 Flowers and Branches - Most flowers express the Fibonacci sequence if you
count the number of petals on these flowers. Some plants also exhibit the
Fibonacci sequence in their growth points, on the places where tree
branches form or split.
 Honeybees - The family tree of a honeybee perfectly resembles the
Fibonacci sequence. A honeybee colony consists of a queen, a few drones
and lots of workers.
 Human body has many elements that show the Fibonacci numbers and the
golden ratio. Most of your body parts follow the Fibonacci sequence and
the proportions and measurements of the human body can also be divided
up in terms of the golden ratio.
 Geography, Weather and Galaxies Fibonacci numbers and the relationships
between these numbers are evident in spiral galaxies, sea wave curves and
in the patterns of stream and drainages.
The Golden Ratio and/or the Golden Spiral can also be observed in music, art,
and designs:
  Architecture - The Great Pyramid of Giza: The Great Pyramid of Giza built
around 2560 BC is one of the earliest examples of the use of the golden
ratio.
- The Greek sculptor Phidias sculpted many things including the bands
of sculpture that run above the columns of the Parthenon.
 Arts - Mona-Lisa by Leonardo Da Vinci: It is believed that Leonardo, as a
mathematician tried to incorporate of mathematics into art.
MATHEMATICS FOR OUR WORLD:
“Neglect of mathematics works injury to all knowledge, since he who is ignorant
of it cannot know the other sciences or the things of the world..” - Roger Bacon
(1214-1294)
 Mathematics is everywhere; whether it is on land, sea or air, online or on
the front line, mathematics underpins every nook and cranny of modern
life.
 Math helps us understand or make sense of the world – and we use the
world to understand math. It is therefore important that we learn math
contents needed to solve complex problems in a complex world
Applications of Mathematics in our world:
 Mathematics helps organize patterns and regularities in the world
 Mathematics helps predict the behavior of nature and many phenomena
 Mathematics helps control nature and occurrences in the world for our
own good
 Mathematics has applications in many human endeavors.

Arithmetic Sequences:
Sequence - is a list of numbers typically with a pattern.
- a set of numbers in a specific order.
Term – each number in the list
- the numbers in the sequence
  The first term in a sequence is denoted as a1, the second term is a2, and so
on up to the nth term an.
Arithmetic sequence – if the difference between successive terms is constant.
Common difference – the difference between the terms

Geometric Sequence:
Geometric Sequence - if the ratios of consecutive terms are the same.

Difference table:
Difference table - shows the differences between successive terms of the
sequence. The differences in rows maybe first, second and third differences. Each
number in the first row of the table is the differences between the closest
numbers just above it. If the first differences are not the same, compute the
successive differences of the first differences.
“The essence of mathematics is not to make simple things complicated, but to
make complicated things simple.” - S. Gudder

Propositions - Mathematics is a language. As in any other types of language, we

use sentences to communicate thoughts and ideas. Mathematics is not an
exception. We use propositions to communicate mathematical ideas precisely.
- Is a declarative sentence what can be objectively identified as either true or
false, but not both. If a preposition is true, then its truth value is TRUE and
is denoted by T or 1; otherwise, its truth value is false and is denoted by F
or 0.
 Symbolically, we denote prepositions in the lessons using lower case letter,
such as p, q, r, s, etc.
Negation – of a preposition p is the preposition which is false when p is true; and
true when p is false. The negation of p is denoted by ¬p.
In the English language, we can simply state the negation of a proposition p by
saying “It is not the case that p.” However, there are many ways to express
negations of statements grammatically by replacing “is/are” by “is not/are not”,
etc.
Example 2. Given the statements.
p : Everyone in Visayas speaks Cebuano.
q : Today is Wednesday
The corresponding negations are:
¬ p : Not everyone in Visayas speaks Cebuano.
¬ q : Today is not Wednesday.
Simple proposition - is a proposition with only one subject and only one predicate.
Example:
the proposition “Every cat that barks has a PhD.” is a simple proposition. The
subject of this proposition is “every cat that barks” and the predicate is “has a
PhD.” In logic, we can combine simple propositions to form compound
propositions using logical connectives. Some of the most common connectives are
“or”, “and”, “but”, “unless”, etc.
Conjunction – of p and q is the proposition “p and q”, denoted by p∧q, which is
true only when both p and q are true.
- In other words, if one of p or q is false, then p ∧ q is false.
p q p∧q
1 1 1
1 0 0
0 1 0
0 0 0

Disjunction – of p and q is the proposition “p or q”, denoted by p V q, which is

false only when both p and q are false.
- In other words, if one of p or q is true (or both), then p ∨ q is true.
p q p∧q
1 1 1
1 0 1
0 1 1
0 0 0

Conditional Statement – p ⟶ q is the proposition “if p, then q.” is the proposition

which is false only when p is true and q is false. The converse, inverse, and
contrapositive of p ⟶ are the conditional statements q ⟶ p, (¬p) ⟶ (¬q), and
(¬q) ⟶ (¬p), respectively.
 In the proposition p ⟶ q, the proposition p is also called as the premise,
hypothesis or antecedent and q is called as the conclusion or consequent.
 A conditional statement is trivially true when the premise is false.
p q p⟶ q
1 1 1
1 0 0
0 1 1
0 0 1

 there are many ways to say p ⟶ q aside from “If p, then q.” Alternatively,
we can say “q if p” or “p implies q”, “p is sufficient for q” or “q is necessary
for p.”
Biconditional Statement – p ⟷ q to be read as “p if and only if q” is the
proposition which is true only if both p and q are true or both p and q are false.
p q p⟷ q
1 1 1
1 0 0
0 1 0
0 0 1

Tautology – if its truth value remains true regardless of the truth values of its
component propositions.
 - if it is always true, no matter what the truth values of the
propositions.
Contradiction – if its truth value remains false regardless of the truth values of its
component proposition.
- if it is always false, no matter what the truth values of the
propositions.
Contingency - A proposition that is neither a tautology nor a contradiction
Set - in mathematics is a collection of well-defined and distinct objects,
considered as an object in its own right. Sets are one of the most fundamental
concepts in mathematics.
Collection - is well-defined if for any given object we can objectively decide
whether it is or is not in the collection. Any object which belongs to a given set is
said to be an element of or a member of the given set.

PROBLEM SOLVING:
Reasoning - is our ability to use logical thinking to produce a decision. There are
two major types of reasoning: inductive and deductive.
Two major types of reasoning:
1. Inductive Reasoning – is the process of reasoning that arrives at a general
conclusion based on the observation of specific examples.
- cannot in general prove general statements as this relies on
examples only
Conjecture – general conclusion
- Could be wrong
- The number of regions in the interior of the circle made by connecting
every pair of points in a set of n points in the circumference is 2n−1.
Counterexamples – can negate our conjectures
2. Deductive Reasoning – is the process of reasoning that arrives at a
conclusion based on previously accepted general statements.
 - is drawing general to specific examples or simply from general case
to specific case. Deductive starts with a general statement (or
hypothesis) and examines to reach a specific conclusions.
- does not rely on examples. We make our conclusion based on
general statements whose truth value is known or assumed. Formal
mathematics is usually based on this type of reasoning
- to prove a certain conjecture.
axioms - basic true statements
theorems – derive true statements from these axioms
George Polya’s Guidelines for Problem Solving:
George Pólya – in 1945, he devised a model for problem solving and published it
in his book How to Solve It. The book contains a collection of mathematical
problems and selected strategies on dealing these. His problem solving model,
which he called heuristic (or serving to discover), is as follows.
POLYA’S FOUR STEPS:
1. Understand the problem - Ask questions, experiment, or otherwise
rephrase the question in your own words.
2. Devise a plan - Find the connection between the data and the unknown.
Look for patterns, relate to a previously solved problem or a known
formula, or simplify the given information to give you an easier problem.
3. Carry OUT the plan - Check the steps as you go.
4. Look back - Examine the solution obtained. In other words, check your
answer
The following are some of his recommended strategies:
1. Draw a diagram.
2. Solve a simpler problem.
3. Make a table.
4. Work backwards.
5. Guess and check.
6. Find a pattern.
7. Use a formula or an equation.
8. Using logical reasoning.